midterm_2_review (dragged) 7

# midterm_2_review (dragged) 7 - d i#2 Make a guess that a...

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MA 36600 MIDTERM #2 REVIEW 7 Say that we can factor Z ( r )= a 0 · p ° k =1 ± r r k ² s k ³ ´µ real roots · q ° k =1 · ± r [ λ k + k ] ²± r [ λ k k ] ² ¸ s k ³ ´µ complex roots . Then the general solution is to the constant coeﬃcient homogeneous equation is y ( t )= p ¹ k =1 º s k ¹ m =1 C km t m 1 » e r k t + q ¹ k =1 ¼º s k ¹ m =1 A km t m 1 » e λ k t cos μ k t + º s k ¹ m =1 B km t m 1 » e λ k t sin μ k t ½ . § 4.3: The Method of Undetermined Coeﬃcients. Consider the constant coeﬃcient diFerential equation a 0 d n y dt n + a 1 d n 1 y dt n 1 + ··· + a n 1 dy dt + a n y = G ( t ) . The Method of Undetermined Coeﬃcients consists of the following steps in order to guess a solution Y = Y ( t ): #1. Express the function on the right-hand side as the sum of functions G ( t )= G 1 ( t )+ G 2 ( t )+ ··· + G r ( t ) where each G i ( t ) is the product of a polynomial, an exponential function, and a trigonometric function. That is, say that we can write G i ( t )= d i ¹ j =0 a ij t j e α i t cos β i t + d i ¹ j =0 b ij t j e α i t sin β i t for some constants a ij , b ij , α i , and β i . Note that G i ( t ) involves a polynomial of degree
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Unformatted text preview: d i . #2. Make a guess that a solution Y i = Y i ( t ) of the nonhomogeneous equation n ¹ m =1 P n − m ( t ) Y ( m ) i ( t ) = G i ( t ) for i = 1 , 2 , . . . , r ; is in the form Y i ( t ) = d i + n ¹ j =0 A ij t j e α i t cos β i t + d i + n ¹ j =0 B ij t j e α i t sin β i t for some constants A ij and B ij . Note that α i , and β i are the same as above, and that Y i ( t ) involves a polynomial of degree ( d i + n ). #3. Recombine as the sum Y ( t ) = Y 1 ( t ) + Y 2 ( t ) + · · · + Y r ( t ); then Y = Y ( t ) is the desired solution to the nonhomogeneous equation a d n Y dt n + a 1 d n − 1 Y dt n − 1 + · · · + a n − 1 dY dt + a n Y = G ( t ) ....
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